There is a definition of $h^{1}(\overline{D})$ by many people, and a definition of $H^{1}(\overline{D})$ by Alexrander Borisov using the notion of ghost spaces of the second kind. But neither is the same as the "original ones" you are talking about.
As Neukirch pointed out in his book (page 210), in the classical setting an analgous definition of $H^{1}(\overline{D})$ is completely missing. The classical way of getting around this issue is to define $\chi(D)$ instead, and to generalize either you sacrifice the exactness of Riemann-Roch or you re-write the theory using $K$- groups. Both have been extensively discussed in Neukirch. As far as I know, apparently the same situation happens for two dimensional Arakelov theory over surfaces as well: we do not have a reasonable definition of $H^{i}$, instead we have a "Faltings volume" that can be used in some ways to form the Euler characteristic. There has been works done by many people to generalize this to higher dimensional situations in 1980s using index theorem, including efforts replacing Faltings volume by Quillen metric and substantial use of characteristic classes.
If we step out of the comfort zone and willing to compromise, then there are "regularization procedure" that allows you to define $h^{1}(\overline{D})$ which essentially boils down to Poisson summation formula, or Pontrajin duality if you prefer. And it has been invented and re-invented by many people. But the essential obstacle is still the same. In particular it seems rather unlikely that we will have an Arakelov cohomology theory built upon derived functors or Cech cohomology alone in the case of arithemetic surface.
The difficulty seems to be concentrated on the particular case of the vertical divisor at infinity, for which the "ghost space approach" collapses and classical approach thrives (see Lang, page 114). In all other cases (horizontal divisor, vertical divisor at finite places) the ghost space gives what we wanted. In particular, it is not clear how to define a united theory that can recover Faltings-Riemann-Roch in two dimensional case. There are quite a few Arakelov theory experts in the forum (Minhyong Kim, for example) who wrote their theses around this topic. I am sure they can provide much more professional feedback.